236 CHEMICAL ENGINEERING and 10, showing the actual development of planes in a test piece. A cube fractured as in Fig. 9 will yield two pyramids, four six-faced figures with triangular faces and eight regular tetrahedrons. After the first crush of the cube the tetrahedral and pyramidal pieces in further reduction in a crushing machine will be caught between the crushing faces in such a way that a face of a fragment will be pressed against one crushing face and a point or edge of a fragment against the other. In pieces of this shape the fracturing planes will radiate from the point or edge of application of pressure. The six-sided figures having a pair of parallel faces will tend to develop conjugate planes in further crushing. In either case there is a tendency to increase in the number of tetrahedrons formed and so there is multiplication of tetrahedrons with successive comminutions. More or less cubical blocks of large size are quite common as the ore or rock comes from the mine or quarry owing to the effect and distribution of the explosive charges in the cleavage planes. The original test cube yields 12 fragments consequently the average volume after the theoretical break is one-twelfth that of the unbroken FIG, 11.—Tetrahedral ore fragments (actual size). cube. All of these fragments require a square aperture of half the width of the original cube just to refuse to pass them or just to pass them. If the original tost cube broke into others of half the length of edge of the original cube there would of course be eight of them and the volume of each would be one-eighth that of the original cube. With successive breaks there is an increase in the number of tetrahedrons formed but the discrepancy in working volume between that of a cube and a right tetrahedron of equal edge is not so great as the figures of the respective volumes would indicate. A right tetrahedron has but 11.78 per cent of the volume of a cube with the same edge. Perfect tetrahedrons with respect to the points are rarely to be found. On this account their percentage volume of the smallest enclosing cube is greater than that of any regular tetrahedron similarly enclosed. Figure 11 shows two ore fragments of the tetrahedral shape. Both reproductions are full size. By careful measurement with sliding wires fragment A has been found just to pass in one direction and one only a square aperture whose side is 1.2 in. The volume of the fragment is 0.6136 cu. in. obtained by water displacement. Fragment B will just pass an aperture of 1.5 in. and its volume is 1.4724 cu. in. The smaller fragment is consequently 35.5 per cent of the volume of the cube with side 1.2 and the volume of the larger 43.6%per cent of the volume of the cube with side of 1.5 in.